Numerical boundaries for some classical Banach spaces

Globevnik gave the definition of boundary for a subspace A ⊂ C_b (Ω). This is a subset of Ω that is a norming set for A. We introduce the concept of numerical boundary. For a Banach space X, a subset B ⊂ Π (X) is a numerical boundary for a subspace A ⊂ C_b (B_X, X) if the numerical radius of f is the supremum of the modulus of all the evaluations of f at B, for every f in A. We give examples of numerical boundaries for the complex spaces X = c₀, C (K) and d_* (w, 1), the predual of the Lorentz sequence space d (w, 1). In all these cases (if K is infinite) we show that there are closed and disjoint numerical boundaries for the space of the functions from B_X to X which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of c₀, we characterize the numerical boundaries for that space of holomorphic functions.